非线性模型GAM与MARS的假设辨析及正态性问题下的数据处理咨询

Great questions—these are common points of confusion when working with flexible nonlinear models. Let’s unpack each one:

1. Do MARS and GAM models assume heteroscedasticity and IID errors? #

The short answer is: it depends on the specific implementation.

• For traditional, OLS-based GAMs (e.g., using cubic splines with least squares fitting), the core assumptions mirror linear regression: IID (independent and identically distributed) errors with constant variance (homoscedasticity). However, modern GAM frameworks (like those in mgcv for R) support extensions that relax these assumptions—think weighted least squares for heteroscedasticity, or generalized estimating equations (GEE) for clustered/non-independent data.
• Base MARS (as introduced by Friedman in 1991) also relies on IID, homoscedastic errors for its fitting procedure. The literature disagreement you’re seeing likely stems from applied extensions: some researchers have adapted MARS with weighted fits or robust loss functions to handle heteroscedasticity, while others stick to the original formulation. So the "assumption" varies based on whether you’re using the vanilla model or a modified version.

2. Is MARS more robust than GAM, even though this isn’t stated in the original paper? #

First, we need to clarify what "robustness" means here—usually resistance to outliers, model misspecification, or noisy data.

• The original MARS paper doesn’t explicitly frame robustness as a key feature. That said, many practitioners have observed that MARS can be more robust in certain scenarios: its piecewise linear structure fits local segments of the data, so a single outlier in one segment is less likely to skew the entire model compared to a GAM’s smooth spline, which spreads influence across a wider range.
• That said, this isn’t a universal rule. GAMs can be made robust too (e.g., using robust regression loss functions instead of least squares), and MARS can still be sensitive to outliers that fall at the knots where segments meet. The "MARS is more robust" claim comes from practical empirical results, not the original theoretical formulation.

3. Should we use Box-Cox or Yeo-Johnson transformations when normality assumptions fail? #

Not necessarily—transformations are one tool, but they’re not always the best solution. Here’s how to approach it:

• If you’re using a generalized GAM (GLM-GAM), you can directly specify a non-normal error distribution (e.g., Poisson for count data, Gamma for positive skewed data, binomial for binary outcomes) instead of transforming the response. This is often preferable because it preserves the original interpretation of the response variable.
• If you’re stuck with a traditional normal-error framework:
  • Box-Cox works well for positive, skewed responses, but fails with zeros or negative values. Yeo-Johnson extends this to handle non-positive data, but both transformations alter the meaning of your coefficients (you’re now modeling the transformed response, not the original).
  • Alternatives to consider: weighted regression to address heteroscedasticity (a common cause of non-normality), robust regression methods that downweight outliers, or nonparametric error estimation.
• The key question is: why is normality failing? If it’s due to inherent skewness in the data, transformation might help. If it’s due to heteroscedasticity or outliers, other methods are likely better.

内容的提问来源于stack exchange，提问作者vcar97